Taylor - Theoretic Arithmetic.pdf - Platonic Philosophy

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Farther still, if 2 be subtracted from the number of the rank which any number after 28, holds in the series 6, 28,496, 8128, 130816, kc. and the remainder be added to the number of the said rank, the sum will be the index of that power of 2 which by multiplication with its corresponding number in the series 3, 7, 31, 127, 511, &c. produced the perfect or partially perfect number. Thus if from 3, we subtract 2, and add 3 to the remainder 1, the sum 4 will indicate that the fourth power of 2, viz. 16 multiplied by the third term 31, will give the third term, viz. 4%, in the series 6, 28, 496, 8128, &c. Thus also, if 2 be subtracted from 4, and 4 be added to the remainder, the sum 6 will indicate that the sixth power of 2, viz. 64, multiplied by the 4th term 127, will produce the 4th term 8128, of the series 6, 28, 496, 8128, &c. And so of the rest. The rule for obtaining a number which is either perfect or partially perfect in the series 6, 28, 496, 8128, 130816, &c. any term in this series being given, in the most expeditious manner, is the following. Multiply the given perfect, or partially perfect number by 16, and add to the product twelve times the number in the duple series, from the multiplication of which with a corresponding number in the series 3, 7,31, 127, 511, &c. the perfect, or partially perfect number is produced, and the sum will be the next number in order in the series 6, 28, 496, &c. Thus 28 x 16=448, and 448 added to 12 times 4, i.e. to 48, is equal to 496. Thus too 496X 16=7936, and 7936 + 12 X 16(=192)=8128. And so of the rest. Lastly, all tllr pcrfect numbers arc found in the series of hexagonal numbers; which numbers are 1, 6, 15, 28, 45, 66, &c.; and the expression which is the aggregate of them and when evolved gives all of them in order ad infinitum is 1+3
BOOK Two CHAPTER XL On another species of imperfectly amicable n urn bers. As Ozanam in his Mathematical Recreations, does not mention what numbers are to be employed in order to find other amicable numbers after 9363584, and 9437056," after various trials to accomplish this, I found that if any number by which two amicable numbers are obtained, is multiplied by 8, the product will be a number by which either two numbers may be obtained, the aggregate of all the parts of one of which is equal to the other, or two numbers the aggregate of the parts of one of which only arising from a division by 2 and its powers, will be equal to the other. Thus if 64 is multiplied by 8, the product will be 512. The three numbers formed in the same way as the primes that produce amicable numbers, will be 1535, 3071, and 4718591. And two numbers 4827120640, 4831837184, will be produced, the sum of the parts of the latter of which arising from a division by 2 and its powers with the addition of unity, will be equal to the former. Again, if 512 is multiplied by 8, the product will be 4096. The three numbers corresponding to primes will be 12287, * "Schooten gives the following practical rule from Descartes, for finding amicable numbers, viz. Assume the number 2, or some power of the number 2, such that if unity be subtracted from each of these three following quantities, viz. from 3 times the assumed number, also from 6 times the assumed number, and from 18 times the square of the assumed number, the three remainders may be all prime numbers; then the last prime number being multiplied by double the assumed number, the product will be one of the amicable numbers sought, and the sum of its aliquot parts will be the other.*'-Hutton's Mathcm. Dict. This also is Ozanam's method of finding amicable numbers, which we have already given and illustrated by examples. Mr. John Gough shows, that if a pair of amicable numbers be divided by their greatest common measure, and the prime divisors of these quotients be severally increased by unity, the products of the two sets thus augmented, will be equal.
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THEORETIC ARITHMETIC IN THREE BOOKS
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IV INTRODUCTION the mathematical di
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INTRODUCTION works, like the remain
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VIII INTRODUCTION fore, that the so
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INTRODUCTION the other of such as a
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XII INTRODUCTION must her inherent
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XIV INTRODUCTION principles existin
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XVI INTRODUCTION indissoluble perma
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XVIII INTRODUCTION mathematics to p
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XX INTRODUCTION utility it administ
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XXII INTRODUCTION may not energize
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INTRODUCTION tradition, however, of
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XXVI INTRODUCTION Nicomachus, which
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INTRODUCTION perplexity, and that h
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INTRODUCTION ever, requisite that w
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XXXII INTRODUCTION that he is ignor
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XXXIV INTRODUCTION tirely the prais
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THEORETIC ARITHMETIC BOOK ONE CHAPT
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that all motion is after rest, and
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since it is not possible for divisi
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dle. And these indeed are the commo
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monad is either even or odd. It can
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is denominated the evenly-odd, and
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terms should accord with its proper
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ural order, 1.3.5.7.9.11.13.15.17.
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agreeably to the form of the evenly
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12, 20, and 28, the sum of the extr
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numbers proceeding from them are re
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THE generation however and origin o
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ever will give the mode of measurin
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THE SIEVE OF ERATOSTHENES. Here 7 m
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pass the sum of the whole number, i
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CHAPTER XV. On the generation of th
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CHAPTER XVI. On relative quantity,
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consequent order, these even number
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For the sesquialter has for its lea
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CHAPTER XIX. That the multiple is m
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which 4 is, is compared with the ro
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number besides containing the whole
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superquadripartient, may likewise b
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to them, duple sesquiquartan ratios
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the impression of itself, defines a
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If, however, the multiples which ar
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tan, the superquadripartient ratio
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BOOK TWO CHAPTER I. How all inequal
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BOOK Two 57 reduced to equality. Fo
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BOOK Two 59 alters as much distant
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BOOK Two 61 ratio is composed from
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BOOK Two CHAPTER IV. On the qztanti
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BOOK Two 65 distended by a triple d
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These triangular numbers are genera
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BOOK Two 69 But in these numbers al
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BOOK Two 71 triangle of a class imm
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BOOK Two 73 22 is formed from the s
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BOOK Two 75 third, a pentagonal; th
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Two does not only not arrive at uni
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BOOK Two 79 to have four angles and
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BOOK Two 81 parity. Hence, in conse
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BOOK Two 83 the principles of thing
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BOOK Two 85 from the monad, are sai
Page 125 and 126: pared with the first number longer
Page 127 and 128: BOOK Two 89 and twice 16, has for i
Page 129 and 130: atios. Let them, therefore, be alte
Page 131 and 132: BOOK Two 93 ity. For 1 is the whole
Page 133 and 134: BOOK Two 95 ticipate of an immutabl
Page 135 and 136: BOOK Two 97 are opposite to those a
Page 137 and 138: BOOK Two is equal to the sum of the
Page 139 and 140: BOOK Two 101 will be the same multi
Page 141 and 142: BOOK Two 103 terms there is a great
Page 143 and 144: BOOK Two 105 from 8, and one side f
Page 145 and 146: BOOK Two 107 by the greater, by the
Page 147 and 148: BOOK Two 109 tertian ratio, or that
Page 149 and 150: BOOK Two 111 ratio, arises from a c
Page 151 and 152: BOOK Two 113 to 10 is a quadruple r
Page 153 and 154: BOOK Two 115 For here the greatest
Page 155 and 156: BOOK Two 117 by length, breadth, an
Page 157 and 158: BOOK Two The greatest harmonies. Pr
Page 159 and 160: BOOK Two 121 which is 18416, we mus
Page 161 and 162: BOOK Two former diameter, exceeds i
Page 163 and 164: BOOK Two In the geometrical fractio
Page 165 and 166: BOOK Two The parts of 51=17)
Page 167 and 168: BOOK Two 129 In the second place, i
Page 169 and 170: BOOK Two them; (see Chap. 15, Book
Page 171: BOOK Two 133 of this series into th
Page 174 and 175: Here it is evident in the first ins
Page 178 and 179: 24575, and 301989887. And the two i
Page 180 and 181: In like manner if 9437056 be divide
Page 182 and 183: These numbers are as follows: In th
Page 184 and 185: That the reader may see the truth o
Page 186 and 187: all perfectly as well as all imperf
Page 188 and 189: parts of these wholes have sometime
Page 190 and 191: these, and are called by modern mat
Page 192 and 193: The numbers, therefore, that form 1
Page 194 and 195: with each other, it will be found,
Page 196 and 197: PYTHAGORAS
Page 198 and 199: ples of these; referring the point
Page 200 and 201: If it be requisite, however, to spe
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Page 204 and 205: that they are separate, they are no
Page 206 and 207: 27, but they signify through these
Page 208 and 209: dissimilarly similar to divinity. I
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Page 212 and 213: It is also Justice, and Isis, Natur
Page 214 and 215: adytum of god-nourished silence, as
Page 216 and 217: called Cupid for the reason above a
Page 218 and 219: the diapente, and the diatessaron.
Page 220 and 221: $Fin of natural effects. Because li
Page 222 and 223: er is not alone even, nor alone odd
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and beauty. And in external things,
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which are productive of life after
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one side, and 1 on the other, each
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The same writer also observes, "tha
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position; for I conjecture this to
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~ c ~ ~ ~ Or ~ Q rather, ~ v . whic
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The ratio of 12 to 9 is sesquiterti
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monad. Thus if from each of the num
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had these appellations in consequen
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and well-ordered digression (from t
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unity is added to the product, the
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And so of the rest, which abundantl
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Again, of any two numbers whatever,
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Again, if 4 multiplies any triangul
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With respect to the hexad, it is th
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and the double of it 12, the number
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incorporeal and corporeal essence;
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of the hebdomad 3 and 4 necessarily
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composition of which the hebomad is
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it is also surpassed by the last nu
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excite the glad husbandmen to the c
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mentioned particulars, but also to
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7X 1 and +1= 8 the 2nd 7 x 8 and +
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8X 1 and +1= 9 the 2ndl 8X 7and +1=
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10X 1 and +1= 11 the 2ndl i' 10x30
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tureX according to this number. In
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through a good allotment, or throug
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titude, and communion with itself,
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evenlyeven numbers, and for not dis
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ing this table from the Epanthematc
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for instance, 12 is a number longer
show all

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